1. Field of the Invention
The present invention relates to a method of determining the volume of an aliquot of a sample solution, and more specifically to a method of determining the volume of an aliquot of a sample solution which involves combining the dye-addition and dye-dilution methods of liquid volume determination, and even more specifically to a method of both precisely and accurately determining the volume of an aliquot of a sample solution.
The present invention also relates to an apparatus for determining the volume of an aliquot of a sample solution, and more specifically to an apparatus for determining the volume of an aliquot of a sample solution which performs part of the method of the present invention, and even more specifically to an apparatus for both precisely and accurately determining the volume of an aliquot of a sample solution.
The present invention also relates to a kit for determining the volume of an aliquot of a sample solution, and more specifically to a kit for determining the volume of an aliquot of a sample solution which involves using the method and apparatus of the present invention, and even more specifically to a kit for both precisely and accurately determining the volume of an aliquot of a sample solution.
Finally, the present invention relates to a method, apparatus, and kit for calibrating a liquid delivery device, such as a pipette, but it is not limited to being used only for that purpose.
2. Description of the Prior Art
Many experimental protocols require use of a small liquid volume, including one of about 5 ml or less. Sometimes this volume must be precisely and accurately known. This is true because in such instances, even a slight error can be deleterious. For example, it is well known in the field of cellular biology that a small discrepancy between the actual and recorded volumes of Bradford reagent used in a small-scale Bradford reaction can significantly skew protein concentration calculation. Where a volume must be precisely and accurately known, it therefore is essential to measure and deliver that volume using a liquid delivery device which has been precisely and accurately calibrated.
Methods and apparatuses which can be used to determine small liquid volumes, and therefore which can be used to calibrate liquid delivery devices, include those which involve spectrophotometric measurement of the light absorbance of a dye-containing liquid sample. For example, U.S. Pat. No. 6,741,365 issued to Curtis (the “Curtis '365 patent”), describes methods and apparatuses for liquid delivery device calibration. The entire contents of the Curtis '365 patent are incorporated herein by reference.
It can be generally stated that there have been only two methods for determining small liquid volumes using spectrophotometry. These methods are referred to as: (1) the dye-addition method, described in the Curtis '365 patent; and (2) the dye-dilution method, which is described in an international standard, ISO 8655 part 7, dated 1 Sep. 2005, which also is incorporated herein by reference. Both of these methods employ a well known relationship entitled the Beer-Lambert Law, according to which the absorbance of light by a dye solution is given by:A=εdCdl  (1)where A is the absorbance (dimensionless) of light by the solution at a chosen wavelength, εd (cm−1 liters/mole) is the extinction coefficient of the dye molecules at that same wavelength (a measure of their ability to absorb light of the wavelength being used for the measurement), Cd (moles/liter) is the concentration of dye molecules in the solution, and l (cm) is the pathlength of light through the solution. Typically, the wavelength of light is chosen to be at or near an absorbance maximum for the dye solution.
According to the dye-addition method, a known volume of diluent solution Vb is put into a measurement vial suitable for making optical measurements. An unknown volume Vsof sample solution containing dye with concentration Cs then is delivered into the known volume of diluent solution. The two solutions are mixed together, and the absorbance of the mixture is measured in a spectrophotometer at a first wavelength λs. The concentration of dye in the resulting mixture is given by:
                    C        =                              C            s                    ⁡                      [                                          V                s                                                              V                  b                                +                                  V                  s                                                      ]                                              (        2        )            From the Beer-Lambert Law and the results of the absorbance measurement, the volume Vs of sample solution that was added is calculated using the formula:
                              V          s                =                              V            b                    ⁡                      [                                          A                s                                                                                  ɛ                    s                                    ⁢                                      c                    s                                    ⁢                  l                                -                                  A                  s                                                      ]                                              (        3        )            In this method, the absorbance of the mixture As is measured at the same wavelength as is the extinction coefficient εs. In its simplest implementation, only one wavelength of measurement needs to be employed for this method. Typically, the wavelength of measurement λs is chosen to be at or near the absorbance maximum of the dye.
A variation of this dye-addition method, which is described in the Curtis '365 patent and in U.S. Pat. No. 5,064,282 issued to Curtis, U.S. Pat. No. 5,298,978 issued to Curtis et al., and U.S. Pat. No. 5,492,673 issued to Curtis et al., and pending U.S. Patent Application No. 2005/0168737 by Bradshaw et al., all of which are incorporated herein by reference, requires measurement at a second wavelength λb. In this variant, a dye having an absorbance maximum at λb is added to the diluent solution, and the concentration Cb of the dye in the diluent solution and the dye's extinction coefficient εb are determined or are otherwise obtained. Before any sample solution is added to the diluent solution, the absorbance of the diluent solution Ab is measured at wavelength λb for the purpose of accurately determining the pathlength l. This marks the only time that absorbance is measured at wavelength λb. When the Beer-Lambert law is applied, the volume of sample solution added is given by:
                              V          s                =                              V            b                    ⁡                      [                                                            A                  s                                /                                  A                  b                                                                              (                                                            ɛ                      s                                        ⁢                                                                  C                        s                                            /                                              ɛ                        b                                                              ⁢                                          C                      b                                                        )                                -                                  (                                                            A                      s                                        /                                          A                      b                                                        )                                                      ]                                              (        4        )            
When additional aliquots of sample solution are added to the same vial, volume Vs(n) of the nth such delivery is given by the relationship:
                                          V            s                    ⁡                      (            n            )                          =                                            V              T                        ⁡                          (                              n                -                1                            )                                ⁡                      [                                                            (                                                                                    A                        s                                            ⁡                                              (                        n                        )                                                              -                                                                  A                        s                                            ⁡                                              (                                                  n                          -                          1                                                )                                                                              )                                /                                  A                                      b                    ⁢                                                                                                                                                              (                                                            ɛ                      s                                        ⁢                                                                  C                        s                                            /                                              ɛ                        b                                                              ⁢                                          C                      b                                                        )                                -                                  (                                                                                    A                        s                                            ⁡                                              (                        n                        )                                                              /                                          A                      b                                                        )                                                      ]                                              (        5        )            In this relationship, VT(n−1) is the total volume of liquid in the vial after the n−1th delivery, and VT(n−1) is obtained by adding all the volume calculation results up to and including Vs(n−1) to the initial volume Vb. As(n) is the absorbance measured at the first wavelength after the nth sample addition.
A significant limitation of the dye-addition method just described is that it tends to be inaccurate when the volume being measured is a significant fraction of the diluent volume Vb (e.g. Vs≧⅕ Vb). Indeed, the dye-addition method typically yields multiple volume values which are of similar size to one another (i.e., they are precise), but it does not always yield volume values which reflect the true volume of the sample being tested (i.e., they are not accurate). Due to this limitation, the dye-addition method is not ideally suited for determining large volumes and therefore it also is not ideally suited for calibrating devices which deliver large volumes.
The reason for this limitation is that exact values of the quantities εs, Cs, εb, and Cb must be determined before sample solution volume can be determined. To the extent that any or all of εs, Cs, εb, and Cb are inexactly known (e.g., due to evaporation or to solution degradation), error will occur in the calculated results. This is especially true in the instance that the volume Vs is an appreciable fraction of the diluent volume Vb, since then the denominator in equation (4) is relatively small (the difference between two larger numbers), and is accordingly sensitive to error in either of the two terms in the denominator.
The dye-addition method also is susceptible to “trending”, which specifically is a phenomenon whereby volume determination error progressively and cumulatively increases as more and more aliquots of sample solution are added to diluent solution. The volume calculation in equation 5 is the product of the previous volume calculation, which contains error, and a second term with a denominator, which also contains error. This multiplicative error propagation leads to data trending as more and more aliquots are added. For example, when using the dye-addition method to determine the volumes of a series of sample solution volumes, one typically will observe a relatively small level of error, for example, 0.2%, in the volume of a first sample, but see that error level climb significantly higher, for example, to 2%, by the time the volume of a tenth delivery is determined. Indeed, error in data generated by using the dye-addition method can reach a level that some applications will not tolerate after only a few additions of sample solution.
The second of the two methods which involve spectrophotometrically measuring the light absorbance of a liquid sample, the dye-dilution method, is less well known; however, it is described in an international standard, ISO 8655 part 7, dated 1 Sep. 2005, which is incorporated herein by reference. The first step of this method is to add a known amount Vb of solution containing a dye with absorbance maximum at wavelength εb to a vial and then measure the initial absorbance Ab(i) at wavelength λb of this dye solution. Next, an aliquot of clear liquid solution (e.g., water or buffer) of volume Vs is added, which effectively dilutes the concentration of dye in the original solution, the two solutions are mixed together and a final absorbance Ab(f) is measured. When the Beer-Lambert Law is applied to this method, the volume of sample solution (clear liquid) which was added is given by:
                              V          s                =                              V            b                    ⁡                      [                                                                                A                    b                                    ⁡                                      (                    i                    )                                                  -                                                      A                    b                                    ⁡                                      (                    f                    )                                                                                                A                  b                                ⁡                                  (                  f                  )                                                      ]                                              (        6        )            If a series of samples are added to the same vial, the calculated results for the nth addition are:
                                          V            s                    ⁡                      (            n            )                          =                              V            b                    ⁡                      [                                                                                A                    b                                    ⁡                                      (                    0                    )                                                                                        A                    b                                    ⁡                                      (                    n                    )                                                              -                                                                    A                    b                                    ⁡                                      (                    0                    )                                                                                        A                    b                                    ⁡                                      (                                          n                      -                      1                                        )                                                                        ]                                              (        7        )            where Ab(0) is the absorbance measured before any (clear) sample solution is added.
While the dye-dilution method as described above is performed using a sample solution which is clear or, at a minimum, has a coefficient of extinction of zero at the wavelength of measurement, it is not a requirement, provided the coefficient of extinction of the sample solution at the wavelength of measurement is known. The mathematics for such a method can be derived by one skilled in the art.
Unlike the dye-addition method, which tends to be precise but not accurate, the dye-dilution method tends to be accurate but not precise. In other words, by using the dye-dilution method alone, a user likely will obtain a mean volume value which approximates the true volume of an aliquot of a sample solution (i.e., the volume is accurate), but the multiple individual volume values obtained to generate the mean likely will not approximate one another (i.e., they will not be precise). The dye-dilution method tends to be more imprecise when it is used to determine volumes much smaller than Vb (e.g., ≦⅕ Vb). Increased imprecision at this reduced volume level is attributable to the high relative error in measuring the (relatively small) absorbance differences between Ab(n) and Ab(n−1). While some applications may tolerate such imprecision, others will not. Therefore, since it tends to be imprecise, the dye-dilution method is not ideally suited for determining small volumes, which further means that it also is not ideally suited for calibrating delivery devices which must precisely deliver small volumes.
In light of the above mentioned limitations of the dye-addition and dye-dilution methods, what is needed, therefore, is a system for both precisely and accurately determining the volume of a liquid sample which can be performed and used to precisely and accurately calibrate a liquid delivery device. The system should include one or more of a determination method, an apparatus, and a kit combining an apparatus and instructions for carrying out the method.